![]() ![]() (From which we use spherical trigonometry to obtain distances and angles, e.g. Looking at the spherical geometry, it’s also easy to see why positively curved Riemannian geometry was the first to be developed – because it was based on the already familiar geometry of the sphere. Thus, the sum of the angles of the triangle formed by 3 geodesics will always total > 180 degrees. 1), the positive (Riemannian) on the left, and the negative (Lobachevskian) on the right.Īs can be ascertained by inspection (looking carefully at the meridian circles and latitude parallels in the Riemannian sphere), there are no two parallel lines in the Euclidean sense, since any two geodesics (curves of shortest path) must intersect(see Fig. ![]() For convenience – the two spaces are depicted separately above (Fig. ![]() Thus did I portray both antimatter and matter in one superspace or super-manifold. In my actual model I employed a transparent sphere for a postively curved - Riemannian geometry and inside it, enfolded within, a negatively –curved opaque space. One of these (matter) had positive curvature and the space-time was Riemannian (after Bernhard Riemann), while the other (anti-matter) had negative curvature and the space time was Lobachevskian (after Janos Bolyai and Nicholai Lobachevsky – who developed it). Thus, each mass-energy component occupied a differently curved space-time, both being non-Euclidean. I proposed the hypothesis that matter and anti-matter universes co-existed not only in a kind of energy equilibrium, but one of coupled space as well. My first exposure to non-Euclidean Geometry occurred in late 1963, as I put the finishing touches on my science fair project's model of a “dual” universe, consisting of matter and antimatter in dynamic equilibrium. ![]()
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